The algebraic structure of the set of doubly stochastic circulants is that of a semi-ring. The concept of a prime in the doubly stochastic circulants is introduced in this paper and examples are given. The classiication of a prime in the doubly stochastic circulants is equivalent to the solvability of a linear equation over a doubly stochastic circulant. A representation of doubly stochastic ci...

Let Tn be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T ∈ Tn chosen uniformly at random in the set Tn. A simple algorithm is presented to allow direct sampling from the uniform distribution on Tn. Using this algorithm, the elements abo...

We introduce a class of preconditioners for general sparsematrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that Bij = Uij 2 for i , j=1, . . . ,N. The set U3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic doubly stochastic matrices. We compute the volume of the set U3 with respect to the flat Lebesgue measure and analytically evaluate ...

A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected d-regular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on van-derWarden’s conjecture (proved independently by Egorychev (1981) and by Falikman (1981))...

I describe a new formalization for computation which is similar to traditional circuit models but which depends upon the choice of a family of [semi]groups – essentially, a choice of the structure group of the universe of the computation. Choosing the symmetric groups results in the reversible version of classical computation; the unitary groups give quantum computation. Other groups can result...

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.

This paper deals with the design and analysis of dynamical systems on directed graphs (digraphs) that achieve weight-balanced and doubly stochastic assignments. Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of coordination problems, including formation control, agreement, and distributed optimization. We refer to a digraph as...

Motivated by the applications in routing in data centers, we study the problem of expressing an n × n doubly stochastic matrix as a linear combination using the smallest number of (sub)permutation matrices. The Birkhoff-von Neumann decomposition theorem proves that there exists such a decomposition, but does not give a representation with the smallest number of permutation matrices. In particul...

Correspondence of features between two or more images obtained from different views of the same object is still a challenging problem in vision. In many applications, these two views are part of a larger video sequence. The aim of this paper is to show that the original video data contains enough information (referred to as prior information) to design a robust correspondence algorithm. A doubl...